First off - welcome to the DIS.
Second - "budget" means different things to different people. I will give you my take on things and you can take it for what it's worth.
I would find a room at the All Stars great for a family of 4, but perhaps a bit crowded for a family of 5. (you need to decide that one for yourself - it's about the size of the adverage HoJo's room.) They are a great value though if you feel the room size is appropriate for you.
There are some nice off site places that have kid's suites and you may be able to get a deal on pricing in late August as many schools are heading back. I prefer to stay onsite - but off site and affordable is better than onsite and stressing over the cash. Research the pricing you can get inside and out for when you want to go. If the pricing is the same or close - go with staying onsite. You more than likely will thank yourself for it.
We prefer to stay at FW. (the campground) We have a camper (just a small popup) and it proves to be the most economical for us. We are able to bring along our small fridge or a cooler - we swing by a local store for some food items and keep our snack and light meal costs down. There are mosquitos in Florida - but just pack some bug spray and you should be fine in the campground. (they are nothing like the blood hungry mosquitos that have attacked me while camping here in PA and NJ!) I have never noticed them in the parks. Other bugs have never been an issue. (Love bugs don't bite, can be a bother, but not a big deal) The heat will be considerable, but if you camp at all or study up a bit, it can be managed. (fans - air conditioner - reflective blankets on the tent ends....some or all of these can help immensely when camping in heat.)
The cabins at FW are very nice, but in my opinion they are just as expensive or more expensive than two rooms at the All Stars. (remember - "bargin" and "budget" are very relative terms.)
I encourage you to stick to your planning as August will be here before you know it. This place is a great place to ask any and all questions.
Good Luck and i hope you enjoy your trip!
- lori